Dirichlet series
| $\zeta_K(s)$ = 1 | + 2·2-s + 3·4-s + 4·8-s + 9-s + 2·13-s + 5·16-s + 17-s + 2·18-s + 2·19-s + 25-s + 4·26-s + 6·32-s + 2·34-s + 3·36-s + 4·38-s + 2·43-s + 2·47-s + 49-s + 2·50-s + 6·52-s + 2·53-s + 2·59-s + 7·64-s + 2·67-s + 3·68-s + 4·72-s + 6·76-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda_K(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(17\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | no |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 17,\ (0, 0:\ ),\ 1)$ |
Euler product
\[\begin{aligned}\zeta_K(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]
Factorization
\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{17}(16, \cdot))\)
Particular Values
\[\zeta_K(1/2) \approx -1.053900426\]
Pole at \(s=1\)